Prison Switcharoo

RAY: I got this from my pal, Stan Zdonik, who teaches at Brown University. It was given to him by a colleague who failed to provide him with the answer -- maybe because he didn't know it?

Here it is.

This puzzle has been making the rounds of Hungarian mathematicians' parties.

The warden meets with 23 new prisoners when they arrive. He tells them, "You may meet today and plan a strategy. But after today, you will be in isolated cells and will have no communication with one another.

"In the prison is a switch room, which contains two light switches labeled A and B, each of which can be in either the on or the off position. I am not telling you their present positions. The switches are not connected to anything.

"After today, from time to time whenever I feel so inclined, I will select one prisoner at random and escort him to the switch room. This prisoner will select one of the two switches and reverse its position. He must move one, but only one of the switches. He can't move both but he can't move none either. Then he'll be led back to his cell.

"No one else will enter the switch room until I lead the next prisoner there, and he'll be instructed to do the same thing. I'm going to choose prisoners at random. I may choose the same guy three times in a row, or I may jump around and come back.

"But, given enough time, everyone will eventually visit the switch room as many times as everyone else. At any time anyone of you may declare to me, 'We have all visited the switch room.'

"If it is true, then you will all be set free. If it is false, and somebody has not yet visited the switch room, you will be fed to the alligators."

Here's the question:

What is the strategy the prisoners devise?

And here are the hints I'll give:

Hint number 1: A sixth grader could figure this puzzler out.

Hint B: Take a long-term perspective.

And Hint III: Solve the puzzler for three prisoners.

TOM: I've got a hint, too. When we were first introduced to this problem, unfortunately, there was a roomful of MIT graduates and we all focused on the fact that 23 was a prime number. Boy, was that stupid!